Angular position and velocity estimation for synchronous machines based on extended rotor flux

ABSTRACT

A method of correcting the determination of extended rotor flux using a lag function and a correction algorithm that closely approximates a pure integrator function to correct for lag function errors that can extend the EFS control down to dynamoelectric machine speeds corresponding to as low as 10 Hz electrical frequency.

FIELD OF THE INVENTION

The invention relates to rotor angular position and velocity sensingsystems for mechanical shaft sensorless control of dynamoelectricmachines, and more particularly to an improved system for resolving theposition of a rotor for a dynamoelectric machine using an estimate ofextended rotor flux.

BACKGROUND OF THE INVENTION

As the aerospace industry moves into the more electric era, invertercontrolled dynamoelectric machine drives become more common onboardaircraft. Next generation dynamoelectric machine controllers must meetmany new system and design challenges including cost reduction andreliability improvement. Shaft sensorless dynamoelectric machine controlholds great promise for meeting these challenges.

An aircraft generator is usable as a motor for engine starting whenpowered by an inverter. To reduce cost and improve reliability, it isvery desirable to eliminate the mechanical shaft sensor for the enginestarter. A novel sensorless synchronous dynamoelectric machine controlbased on dynamoelectric machine flux estimation, designated ExtendedFlux Sensorless (EFS) position sensing, is disclosed in U.S. Pat. No.7,072,790 to Markunas et al. and hereby incorporated by reference.Markunas et al. defines extended rotor flux, which aligns with the rotorfield flux axis. The dynamoelectric machine rotor position and speed areestimated from the extended rotor flux, which is derived fromdynamoelectric machine terminal electrical potential and currentmeasurements (FIG. 1), expressed in the α-β two-axis stationaryreference frame well known in the electric machine technical community.Ideally, a pure integrator is required to reconstruct the flux. However,in practice, a pure integrator suffers from direct current (DC)drifting, initial value holding, and even stability problems. Markunaset al. proposes to alleviate these problems with a lag approximation toan integrator. A digital phase lock loop (FIG. 2) determines the rotorposition and speed from the extended rotor flux. The lag approximationto a pure integrator becomes better at high rotor speeds, asymptoticallyapproaching the characteristics of the integrator at very high speeds.However, at low dynamoelectric machine speeds or equivalently lowsynchronous electrical frequency, ω_(e), the error due to the lagapproximation can become unacceptably large.

SUMMARY OF THE INVENTION

The invention comprises a method of correcting the determination ofextended rotor flux using a lag function and a correction algorithm thatclosely approximates a pure integrator function to correct for lagfunction errors that can extend the EFS control down to dynamoelectricmachine speeds corresponding to as low as 10 Hz electrical frequency.

For a system that derives an estimated rotor electrical position{circumflex over (θ)}_(r) and electrical frequency {circumflex over(ω)}_(r) for the rotor of a polyphase alternating current (AC)dynamoelectric machine with an extended rotor flux estimationcalculation system that generates uncorrected estimated values ofextended rotor flux {circumflex over (λ)}_(α ext unc) and {circumflexover (λ)}_(β ext unc) based on derived values of estimated stator flux{circumflex over (λ)}_(α) and {circumflex over (λ)}_(β) the α-β two-axisstationary reference frame using a lag function

$\frac{1}{s + \omega_{i}}$that approximates a pure integrator function

$\frac{1}{s},$wherein ω_(i) represents a corner frequency of the lag function, and adigital phase lock loop (PLL) to determine values of estimated rotorelectrical position {circumflex over (θ)}_(r) and electrical frequency{circumflex over (ω)}_(r) based upon the estimated values of extendedrotor flux {circumflex over (λ)}_(α ext) and {circumflex over(λ)}_(β ext), one possible embodiment of the invention comprises amethod of correcting the estimated α-axis and β-axis values of estimatedextended rotor flux to more closely approximate values generated with apure integrator function

$\frac{1}{s}$comprising the steps of: dividing the lag function

$\frac{1}{s + \omega_{i}}$corner frequency ω_(i) by the estimated rotary frequency {circumflexover (ω)}_(r) to generate a signal ω_(i)/{circumflex over (ω)}_(r);multiplying the signal ω_(i)/{circumflex over (ω)}_(r) by the derivedvalues of estimated stator flux {circumflex over (λ)}_(α) and{circumflex over (λ)}_(β) to produce signals (ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(α) and (ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(β); adding signal (ω_(i)/{circumflexover (ω)}_(r)){circumflex over (λ)}_(β) to {circumflex over(λ)}_(α ext unc) as represented by {circumflex over(λ)}_(α)−(i_(α)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(α)−(i_(α)×{circumflex over(L)}_(q))+(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β)that represents a corrected value of estimated extended rotor flux{circumflex over (λ)}_(λ ext); and subtracting signal (ω_(i)/{circumflexover (ω)}_(r)){circumflex over (λ)}_(α) from {circumflex over(λ)}_(β ext unc) as represented by {circumflex over(λ)}_(β)−(i_(β)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(β)−(i_(β)×{circumflex over(L)}_(q))−(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(α)representing {circumflex over (λ)}_(β ext).

DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of operations performed within a controllerfor controlling a dynamoelectric machine that calculates extended rotorflux according to the prior art.

FIG. 2 is a block diagram of elements within a controller forcontrolling a dynamoelectric machine that estimates rotor angularposition and velocity of the dynamoelectric machine using a phase lockloop (PLL) according to the prior art.

FIG. 3 is a phasor diagram that shows extended flux sensorlesscorrection according to the invention.

FIG. 4 is an estimated flux linkage triangle that shows extended fluxlinkage according to the invention.

FIG. 5 is an estimated flux linkage similar triangle according to theinvention.

FIG. 6 is an estimated flux sensorless block diagram with correctionalgorithm according to the invention.

DETAILED DESCRIPTION OF THE INVENTION

For Extended Flux Sensorless (EFS) control of a synchronousdynamoelectric machine, the following differential equations estimatethe stator flux linkages in the α-β two-axis stationary reference frameusing a lag approximation to a pure integrator:

$\begin{matrix}{\frac{\mathbb{d}{\hat{\lambda}}_{\alpha}}{\mathbb{d}t} = {v_{\alpha} - {\hat{R_{s}} \times i_{\alpha}} - {\omega_{efs} \times {\hat{\lambda}}_{\alpha}}}} & (1) \\{\frac{\mathbb{d}{\hat{\lambda}}_{\beta}}{\mathbb{d}t} = {v_{\beta} - {\hat{R_{s}} \times i_{\beta}} - {\omega_{efs} \times {\hat{\lambda}}_{\beta}}}} & (2)\end{matrix}$where,

-   -   λ_(α)=α-axis stator flux estimate: V-sec    -   λ_(β)=β-axis stator flux estimate; V-sec    -   v_(α)=α-axis stator potential; V    -   v_(β)=β-axis stator potential; V    -   i_(α)=α-axis stator current; A    -   i_(β)=β-axis stator current; A    -   {circumflex over (R)}_(s)=estimated stator resistance; Ohm    -   ω_(i)=lag function corner frequency; sec⁻¹

For alternating current (AC) steady state analysis, the transformationof these differential equations into Laplace transform notation is:

$\begin{matrix}{{\hat{\lambda}}_{\alpha} = \frac{v_{\alpha} - {{\hat{R}}_{s} \times i_{\alpha}}}{s + \omega_{i}}} & (3) \\{{\hat{\lambda}}_{\beta} = \frac{v_{\beta} - {{\hat{R}}_{s} \times i_{\beta}}}{s + \omega_{i}}} & (4)\end{matrix}$wheres=Laplace operator; sec⁻¹

The magnitudes of these two stator flux linkage estimates at anestimated electrical frequency of {circumflex over (ω)}_(r) are:

$\begin{matrix}{{{\hat{\lambda}}_{\alpha}} = \frac{{v_{\alpha} - {{\hat{R}}_{s} \times i_{\alpha}}}}{\sqrt{{\hat{\omega}}_{r}^{2} + \omega_{i}^{2}}}} & (5) \\{{{\hat{\lambda}}_{\beta}} = \frac{{v_{\beta} - {{\hat{R}}_{s} \times i_{\beta}}}}{\sqrt{{\hat{\omega}}_{r}^{2} + \omega_{i}^{2}}}} & (6)\end{matrix}$and the angles are:

$\begin{matrix}{{\angle{\hat{\lambda}}_{\alpha}} = {{\angle\left( {v_{\alpha\;} - {{\hat{R}}_{s} \times i_{\alpha}}} \right)} - {\tan^{- 1}\left( \frac{{\hat{\omega}}_{r}}{\omega_{i}} \right)}}} & (7) \\{{\angle{\hat{\lambda}}_{\beta}} = {{\angle\left( {v_{\beta\;} - {{\hat{R}}_{s} \times i_{\beta}}} \right)} - {\tan^{- 1}\left( \frac{{\hat{\omega}}_{r}}{\omega_{i}} \right)}}} & (8)\end{matrix}$

One convenient means for deriving the estimated rotor electricalfrequency {circumflex over (ω)}_(r) is the phase locked loop describedin Markunas et al. and shown in FIG. 2. Note that use of the phaselocked loop forces the estimated rotor electrical frequency {circumflexover (ω)}_(r) to be equal to the synchronous electrical frequency ω_(e)of the terminal electrical potential applied to the dynamoelectricmachine.

As compared to the results for a lag approximation, the amplitudes andangles for stator flux estimates determined using a pure integrator are:

$\begin{matrix}{{{\hat{\lambda}}_{\alpha}} = \frac{{v_{\alpha} - {{\hat{R}}_{s} \times i_{\alpha}}}}{{\hat{\omega}}_{r}}} & (9) \\{{\angle{\hat{\lambda}}_{\beta}} = \frac{{v_{\beta\;} - {{\hat{R}}_{s} \times i_{\beta}}}}{{\hat{\omega}}_{r}}} & (10) \\{{\angle{\hat{\lambda}}_{\alpha}} = {{\angle\left( {v_{\alpha\;} - {{\hat{R}}_{s} \times i_{\alpha}}} \right)} - \frac{\pi}{2}}} & (11) \\{{\angle{\hat{\lambda}}_{\beta}} = {{\angle\left( {v_{\beta\;} - {{\hat{R}}_{s} \times i_{\beta}}} \right)} - \frac{\pi}{2}}} & (12)\end{matrix}$

FIG. 3 shows equations (5) through (12) graphically in phasor format.Phasors 2 and 4 represent net electrical potential phasors,v_(α)−{circumflex over (R)}_(s)×i_(α) and v_(β)−{circumflex over(R)}_(s)×i_(β), respectively. FIG. 3 arbitrarily aligns the α-axis netelectrical potential phasor 2 with the y-axis. The alignment of theβ-axis net electrical potential phasor 4 must then extend along thex-axis for a positive sequence 3-phase electrical potential waveformapplied to the synchronous dynamoelectric machine terminals. In the α-βtwo-axis stationary reference frame, α-axis quantities lead β-axisquantities in time for positive sequence. Phasors 6 and 8 represent theestimated stator flux linkages {circumflex over (λ)}_(α) and {circumflexover (λ)}_(β), respectively. Phasors 6 and 8 lag their respective netelectrical potential phasors 2 and 4 by less than 90°. Phasors 10 and 12represent estimated stator flux linkages determined using a pureintegrator and do indeed lag their respective net electrical potentialphasors 2 and 4 by exactly 90°. Phasors 10 and 12 also represent“corrected” estimated stator flux linkages {circumflex over (λ)}_(α) and{circumflex over (λ)}_(β), respectively, as the correction algorithmdisclosed herein results in nominally zero difference between the statorflux linkage estimate determined using a pure integrator and that usinga lag approximation corrected by the scheme being disclosed herein.Phasors 14 and 16 represent phasors generated by the correctionalgorithm that vectorially add to phasors 4 and 6 to create phasors 10and 12, as described herein.

FIG. 4 shows a right triangle formed by phasors 6, 10 and 14 from FIG.3. Phasor 10 represents the hypotenuse of the right triangle, and it isproportional in length to the magnitude of the α-axis stator fluxlinkage estimated using a pure integrator. Phasor 6 represent theadjacent side of the right triangle, and it is proportional in length tothe magnitude of the α-axis stator flux linkage estimated using a lagapproximation. Since the magnitude of phasor 10 is proportional to1/{circumflex over (ω)}_(r) and the magnitude of phasor 6 isproportional to 1/√{square root over ({circumflex over (ω)}_(r) ²+ω_(i)²)}, the magnitude of phasor 14 is necessarily proportional to

$\left( {\omega_{i}/{\hat{\omega}}_{r}} \right){\sqrt{{\hat{\omega}}_{r}^{2} + \omega_{i}^{2}}.}$

FIG. 5 shows a similar right triangle formed by phasors 6, 10 and 14 anddividing the magnitude of each of the sides by √{square root over({circumflex over (ω)}_(r) ²+ω_(i) ²)}. As can be seen the adjacent sideof the right triangle represented by phasor 6 now has a length of unityand the hypotenuse of the triangle represented by phasor 10 is formed byvectorially adding {circumflex over (λ)}_(α) and (ω_(i)/{circumflex over(ω)}_(r))×{circumflex over (λ)}_(β). The mathematical expression for thecorrected α-axis extended flux linkage is:{circumflex over (λ)}_(α corrected)={circumflex over(λ)}_(α)+(ω_(i)/{circumflex over (ω)}_(r))×{circumflex over(λ)}_(β)  (13)and similarly, the mathematical expression for the corrected β-axisextended flux linkage is:{circumflex over (λ)}_(β corrected)={circumflex over(λ)}_(β)−(ω_(i)/{circumflex over (ω)}_(r))×{circumflex over(λ)}_(α)  (14)

The extended flux linkages can be determined with these correctedestimates of stator flux linkages as shown in FIG. 1 and passed to thephase-locked-loop shown in FIG. 2. FIG. 6 shows an overall block diagramof an EFS system 18 according to the invention with the correctionalgorithm included. As described in Markunas et al., a special lagfunction

$\frac{1}{s + \omega_{i}}$substitutes for the pure integrator

$\frac{1}{s},$wherein ω_(i) is a corner frequency of the lag function. An {circumflexover (R)}_(s) function 20 representing estimated stator resistancemultiplies α-axis current on a signal path 22 by {circumflex over(R)}_(s) to produce a signal {circumflex over (R)}_(s)×i_(α) on a signalpath 24. A summer 26 subtracts {circumflex over (R)}×i_(α) on the signalpath 24 from the α-axis potential v_(α) on a signal path 28 to produce asignal v_(α)−({circumflex over (R)}_(s)×i_(α)) on a signal path 30. A

$\frac{1}{s + \omega_{i}}$lag function 32 as described above multiplies v_(α)−({circumflex over(R)}_(s)×i_(α)) on the signal path 30 by

$\frac{1}{s + \omega_{i}}$to produce a signal

$\frac{1}{s + \omega_{i}}\left( {V_{\alpha} + \left( {{\hat{R}}_{s} \times i_{\alpha}} \right)} \right)$on a signal path 34 that represents estimated α-axis stator flux{circumflex over (λ)}_(α). An {circumflex over (L)}_(q) function 36representing estimated q-axis inductance of the dynamoelectric machinemultiples the α-axis current i_(α) on the signal path 20 by {circumflexover (L)}_(a) to produce a signal i_(α)×{circumflex over (L)}_(q) on asignal path 38. Another summer 40 subtracts i_(α)×{circumflex over(L)}_(q) on the signal path 38 from {circumflex over (λ)}_(α) on thesignal path 34 to produce an uncorrected estimated α-axis extended rotorflux signal {circumflex over (λ)}_(α ext unc) as represented by{circumflex over (λ)}_(α)−(−i_(α)×{circumflex over (L)}_(q)) on a signalpath 42.

Uncorrected estimated β-axis extended rotor flux {circumflex over(λ)}_(β ext unc) is determined in a similar way. An {circumflex over(R)}_(s) function 44 representing estimated stator resistance multipliesβ-axis current on a signal path 46 by {circumflex over (R)}_(s) toproduce a signal {circumflex over (R)}_(s)×i_(β) on a signal path 48.Another summer 50 subtracts {circumflex over (R)}_(s)×i_(β) on thesignal path 48 from β-axis potential v_(β) on a signal path 52 toproduce a signal v_(β)−({circumflex over (R)}_(s)×i_(β)) on a signalpath 54.

Another

$\frac{1}{s + \omega_{i}}$lag function 56 multiplies v_(β)−({circumflex over (R)}_(s)×i_(β)) onthe signal path 54 by

$\frac{1}{s + \omega_{i}}$to produce a signal

$\frac{1}{s + \omega_{i}}\left( {V_{\beta} + \left( {{\hat{R}}_{s} \times i_{\beta}} \right)} \right)$on a signal path 58 that represents estimated β-axis stator flux{circumflex over (λ)}_(β). Another {circumflex over (L)}_(q) function 60representing estimated q-axis inductance of the dynamoelectric machinemultiplies the β-axis current i_(α) on the signal path 46 by {circumflexover (L)}_(q) to produce a signal i_(β)×{circumflex over (L)}_(q) on asignal path 62. Another summer 64 subtracts i_(β)×{circumflex over(L)}_(q) on the signal path 62 from {circumflex over (λ)}_(β) on thesignal path 58 to produce an uncorrected estimated β-axis extended rotorflux signal {circumflex over (λ)}_(β ext unc) as represented byλ_(β)−(i_(β)×{circumflex over (L)}_(q)) on a signal path 66. Theseoperations together comprise an uncorrected extended rotor fluxestimation calculation system 68 according to the prior art as describedin Markunas et al. and shown in FIG. 1.

According to the invention, an estimated extended rotor flux correctionalgorithm 70 corrects deviations of {circumflex over (λ)}_(α ext unc)and {circumflex over (λ)}_(β ext unc) as described above to generatecorrected values of estimated extended rotor flux {circumflex over(λ)}_(α ext) for the α-axis and {circumflex over (λ)}_(β ext) for theβ-axis that closely approximate values that would result from operationswith a pure integrator

$\frac{1}{s}.$A digital PLL 72 as described in Markunas et al. and shown in FIG. 2derives estimated rotor electrical frequency {circumflex over (ω)}_(r)and estimated rotor electrical position {circumflex over (θ)}_(r) of thedynamoelectric machine rotor from the corrected estimated extended rotorflux {circumflex over (λ)}_(α ext) for the α-axis and {circumflex over(λ)}_(β ext) for the β-axis that the correction algorithm 70 generates.

The description of the correction algorithm 70 is as follows. Acomparator 74 compares a signal representing the estimated rotorelectrical frequency {circumflex over (ω)}_(r) on a signal line 76 witha signal representing a minimum frequency ω_(o) on a signal path 78 toavoid dividing by zero or overflowing the resulting ratio,ω_(i)/{circumflex over (ω)}_(r), in fixed point math processors. Thecomparator 74 selects the higher level of these two signals to produce asignal representing a selected estimated rotor electrical frequency{circumflex over (ω)}_(r) on a signal path 80. A divider function 82divides the lag function corner frequency {circumflex over (ω)}_(i) on asignal path 84 by the selected estimated rotor electrical frequency{circumflex over (ω)}_(r) to produce a signal ω_(i)/{circumflex over(ω)}_(r) on a signal path 86.

A multiplier 88 multiplies the signal ω_(i)/{circumflex over (ω)}_(r) onthe signal path 86 by the signal {circumflex over (λ)}_(β) on the signalpath 58 to produce a signal (ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β) on a signal path 90. A summer 92 adds theuncorrected estimated α-axis extended rotor flux signal {circumflex over(λ)}_(α ext unc) as represented by {circumflex over(λ)}_(α)−i_(α)×{circumflex over (L)}_(q) on the signal path 42 to thesignal (ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β) on thesignal path 90 to produce a signal {circumflex over(λ)}_(α)−i_(α)×{circumflex over (L)}_(q)+(ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(β) representing {circumflex over(λ)}_(α ext) on a signal path 94.

Similarly, a multiplier 96 multiplies the signal ω_(i)/{circumflex over(ω)}_(r) on the signal path 86 by the signal {circumflex over (λ)}_(α)on the signal path 34 to produce a signal (ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(α) on a signal path 98. A summer 100subtracts the signal (ω_(i)/{circumflex over (ω)}_(r)){circumflex over(λ)}_(α) on the signal path 98 from the uncorrected estimated β-axisextended rotor flux signal {circumflex over (λ)}_(β ext unc) asrepresented by {circumflex over (λ)}_(β)−i_(β)×{circumflex over (L)}_(q)on the signal path 66 to produce a signal {circumflex over(λ)}_(β)−i_(β)×{circumflex over (L)}_(q)−(ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(α) representing {circumflex over(λ)}_(β ext) on a signal path 102.

The signal {circumflex over (λ)}_(α)−i_(α)×{circumflex over(L)}_(q)+(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β)representing {circumflex over (λ)}_(α ext) on the signal path 94 and thesignal {circumflex over (λ)}_(β)−i_(β)×{circumflex over(L)}_(q)−(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(α)representing {circumflex over (λ)}_(β ext) on a signal path 102 serve asinputs to the digital PLL 72. A multiplier 104 multiplies the correctedestimated α-axis extended rotor flux {circumflex over (λ)}_(α ext) witha feedback signal on a signal path 106 from a sine function 108 toproduce an α-axis multiplier output signal on a signal path 110.Likewise, a multiplier 112 multiplies the corrected estimated β-axisextended rotor flux {circumflex over (λ)}_(β ext) with a feedback signalon a signal path 114 from a cosine function 116 to produce a β-axismultiplier output signal on a signal path 118.

A summer 120 subtracts the α-axis multiplier output signal on the signalpath 110 from the β-axis multiplier output signal on a signal path 118to produce a difference signal on a signal path 122. A proportional plusintegral regulator (Pl) function 124 multiplies the difference signal onthe signal path 122 by the function

$K_{p} + \frac{K_{i}}{s}$to produce a Pl output signal on a signal path 126, wherein K_(p) andK_(i) are the proportional and integral gains of the Pl function 124,respectively.

An integral function 128 multiplies the Pl output signal on the signalpath 126 by the function

$\frac{1}{s}$to produce an integration output signal on a signal path 130. Theintegration output signal on the signal path 130 serves as the inputsignal for both the sine function 108 and the cosine function 116 toprovide the PLL. The integration output signal on the signal path 130therefore represents the estimated rotor electrical position θ_(r).

A low pass filter (LPF) function 132 multiplies the Pl output signal onthe signal path 126 by the function

$\frac{\omega_{c}}{s + \omega_{c}},$where ω_(c) is the corner frequency of the LPF function 132 to producethe estimated rotor electrical angular velocity {circumflex over(ω)}_(r) on the signal path 76. The LPF function 132 is desirable tobetter attain a smooth signal for the estimated rotor electricalfrequency {circumflex over (ω)}_(r).

The corrected EFS control algorithm is only effective above some minimumdynamoelectric machine fundamental electrical frequency. Below thisfrequency, the correction algorithm described in equations (13) and (14)becomes inaccurate due to sensor inaccuracies, digital word lengtheffects, and measurement and computation noise. Test results to datewere able to demonstrate acceptable accuracy above an approximate 10 Hzelectrical frequency. Below this threshold speed, determination of rotorposition for inverter commutation requires the use of alternative means.

Described above is a method of correcting the determination of extendedrotor flux using a lag function and a correction algorithm that closelyapproximates a pure integrator function to correct for lag functionerrors. The described embodiment of the invention is only anillustrative implementation of the invention wherein changes andsubstitutions of the various parts and arrangement thereof are withinthe scope of the invention as set forth in the attached claims.

1. For a system that derives an estimated rotor electrical position{circumflex over (θ)}_(r) and electrical frequency {circumflex over(ω)}_(r) for the rotor of a polyphase alternating current (AC)dynamoelectric machine with an extended rotor flux estimationcalculation system that generates uncorrected estimated values ofextended rotor flux {circumflex over (λ)}_(α ext unc) and {circumflexover (λ)}_(β ext unc) based on derived values of estimated stator flux{circumflex over (λ)}_(α) and {circumflex over (λ)}_(β) the α-β two-axisstationary reference frame using a lag function$\frac{1}{s + \omega_{i}}$ that approximates a pure integrator function$\frac{1}{s},$ wherein ω_(i) represents a corner frequency of the lagfunction, and a digital phase lock loop (PLL) to determine values ofestimated rotor electrical position {circumflex over (θ)}_(r) andelectrical frequency {circumflex over (ω)}_(r) based upon the estimatedvalues of extended rotor flux {circumflex over (λ)}_(α ext) and{circumflex over (λ)}_(β ext), a method of correcting the estimatedα-axis and β-axis values of estimated extended rotor flux to moreclosely approximate values generated with a pure integrator function$\frac{1}{s},$ comprising the steps of: dividing the lag function$\frac{1}{s + \omega_{i}}$ corner frequency ω_(i) by the estimatedrotary frequency {circumflex over (ω)}_(r) to generate a signalω_(i)/{circumflex over (ω)}_(r); multiplying the signalω_(i)/{circumflex over (ω)}_(r) by the derived values of estimatedstator flux {circumflex over (λ)}_(α) and {circumflex over (λ)}_(β) toproduce signals (ω_(i)/{circumflex over (ω)}_(r)){circumflex over(λ)}_(α) and (ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β);adding signal (ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β)to {circumflex over (λ)}_(α ext unc) as represented by {circumflex over(λ)}_(α)−(i_(α)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(α)(i_(α)×{circumflex over(L)}_(q))+(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β)that represents a corrected value of estimated extended rotor flux{circumflex over (λ)}_(α ext); and subtracting signal (ω_(i)/{circumflexover (ω)}_(r)){circumflex over (λ)}_(α) from {circumflex over(λ)}_(β ext unc) as represented by {circumflex over(λ)}_(β)−(i_(β)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(β)−(i_(β)×{circumflex over(L)}_(q))−(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(α)representing {circumflex over (λ)}_(β ext).
 2. The method of claim 1,further comprising the step of selecting a value for the estimated rotorelectrical frequency {circumflex over (ω)}_(r) that is at least aminimum frequency ω_(o) to prevent the step of dividing from dividing byzero or overflowing the resulting ratio, ω_(i)/{circumflex over(ω)}_(r).
 3. The method of claim 2, wherein the step of selectingcomprises the steps of: comparing a value of {circumflex over (ω)}_(r)generated by the digital PLL with the minimum frequency ω_(o); andselecting the higher one of the value of {circumflex over (ω)}_(r)generated by the digital PLL and the value of the minimum frequencyω_(o) as {circumflex over (ω)}_(r) for the step of dividing.
 4. For asystem that derives an estimated rotor electrical position {circumflexover (θ)}_(r) and electrical frequency {circumflex over (ω)}_(r) for therotor of a polyphase alternating current (AC) dynamoelectric machinewith an extended rotor flux estimation calculation system that generatesuncorrected estimated values of extended rotor flux {circumflex over(λ)}_(α ext unc) and {circumflex over (λ)}_(β ext unc) based on derivedvalues of estimated stator flux {circumflex over (λ)}_(α) and{circumflex over (λ)}_(β) the α-β two-axis stationary reference frameusing a lag function $\frac{1}{s + \omega_{i}}$ that approximates a pureintegrator function $\frac{1}{s},$ wherein ω_(i) represents a cornerfrequency of the lag function, and a digital phase lock loop (PLL) todetermine values of estimated rotor electrical position {circumflex over(θ)}_(r) and electrical frequency {circumflex over (ω)}_(r) based uponthe estimated values of extended rotor flux {circumflex over(λ)}_(α ext) and {circumflex over (λ)}_(β ext), apparatus for correctingthe estimated α-axis and β-axis values of estimated extended rotor fluxto more closely approximate values generated with a pure integratorfunction $\frac{1}{s},$ comprising: a divider for dividing the lagfunction $\frac{1}{s + \omega_{i}}$ corner frequency ω_(i) by theestimated rotor electrical frequency {circumflex over (ω)}_(r) togenerate a signal ω_(i)/{circumflex over (ω)}_(r); a multipliermultiplying the signal ω_(i)/{circumflex over (ω)}_(r) by the derivedvalues of estimated stator flux {circumflex over (λ)}_(α) and{circumflex over (λ)}_(β) to produce signals (ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(α) and (ω_(i)/{circumflex over(ω)}_(r)){circumflex over (λ)}_(β); a first summer for adding signal(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β) to{circumflex over (λ)}_(α ext unc) as represented by {circumflex over(λ)}_(α)−(i_(α)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(α)−(i_(α)×{circumflex over(L)}_(q))+(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(β)that represents a corrected value of estimated extended rotor flux{circumflex over (λ)}_(α ext); and a second summer for subtractingsignal (ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(α) from{circumflex over (λ)}_(β ext unc) as represented by {circumflex over(λ)}_(β)−(i_(β)×{circumflex over (L)}_(q)) to produce a signal{circumflex over (λ)}_(β)−(i_(β)×{circumflex over(L)}_(q))−(ω_(i)/{circumflex over (ω)}_(r)){circumflex over (λ)}_(α)representing {circumflex over (λ)}_(β ext).
 5. The system of claim 4,further comprising a selector that selects a value for the estimatedrotor electrical frequency {circumflex over (ω)}_(r) that is at least aminimum frequency ω_(o) to prevent the divider from dividing by zero oroverflowing the resulting ratio, ω_(i)/{circumflex over (ω)}_(r).
 6. Thesystem of claim 5, wherein the selector comprises a comparator thatcompares a value of {circumflex over (ω)}_(r) generated by the digitalPLL with the minimum frequency ω_(o); and selects the higher one of thevalue of {circumflex over (ω)}_(r) generated by the digital PLL and thevalue of the minimum frequency ω_(o) as {circumflex over (ω)}_(r) forthe divider.